Question

Write a program that adds all numbers from 2 to 10,000 to a list. Then remove the multiples of 2 (but not 2), multiples of 3 (but not 3), and so on, up to the multiples of 100 . Print the remaining values.

Short answer

Use a list and iterative filter for numbers 2 to 10,000, eliminating multiples of numbers 2 to 100, keeping only self-multiples.

Step-by-step solution

Initialize the List

Create a list containing all numbers from 2 to 10,000. This will include both even and odd numbers, and we will start the process by checking the range for the initial setup.

Remove Multiples Iteratively

Use a loop to iterate over each integer from 2 up to 100. In each iteration of the loop, filter out the multiples of the current integer from the list, except for the integer itself. Implement this by using list comprehension or a filter method where you keep an element in the list only if it is not a multiple of the current integer, or if it equals the current integer.

Define the Filtering Logic

During each iteration, apply a filtering logic to create a new list of numbers. For an integer `i`, keep numbers less than i same or leave the number `i` itself, and remove all other multiples of `i`. This ensures the list gets updated after every iteration, only with numbers that are either prime or remain after the filtering phase.

Print the Results

After completing the filtering loop, print the remaining contents of the list. These are numbers that aren't multiples of any number from 2 through 100, except themselves, and are effectively the numbers that survive all iterations of sieving.

Key concepts

List Manipulation

In Python programming, working with lists is central to many applications and problem-solving scenarios. Lists are versatile data structures that allow you to store collections of items like numbers, strings, or even other lists. To create a list in Python, you can simply write a sequence of items within square brackets, separated by commas, like this: `[2, 3, 4, ..., 10000]`.

Once you have your list, Python provides a range of operations to manipulate it. You can add items using methods like `append()` for single elements or `extend()` for multiple items. Removing unwanted items can be achieved through functions such as `remove()` or by using list comprehensions.

In the context of our exercise, list manipulation is key. Initially, you populate a list with sequential numbers from 2 to 10,000. Then, you iteratively alter this list by removing specific multiples. This approach not only helps solidify understanding of list operations, but also applies these in a practical problem-solving environment.

Algorithm Design

Algorithm design refers to the process of creating a step-by-step solution to a problem. In the exercise task, you must devise a methodical way to efficiently sift through numbers and filter out specific multiples.

The core steps for designing the algorithm in this scenario are as follows:

• Initialization: Start with a comprehensive list of numbers from 2 to 10,000.
• Iteration: Loop through integers from 2 to 100. In each iteration, utilize list comprehensions to filter out unwanted multiples.
• Output: After the filtering is complete, print the resulting list which contains numbers that aren't divisible by any number from 2 to 100, except themselves.

This algorithm exemplifies an efficient way to use loops and conditions to process large datasets. It demonstrates how combining simple iterations can solve complex problems.

Filtering Multiples

Filtering involves removing certain elements from a collection based on specific criteria. In this exercise, you utilize filtering to remove numbers which are multiples of integers ranging from 2 to 100, except for the integer itself.

This is achieved using Python's list comprehensions. A list comprehension provides a concise way to construct lists. It allows you to specify an operation to perform on each list element and a condition to select only certain elements.

For example, in each iteration step, elements remaining in the list after filtering are those that meet the condition: they are not multiples of a current integer, or they are the integer itself: `[x for x in lst if x == i or x % i != 0]`. This succinct syntax is both readable and powerful, making it a preferred technique in Python programming.

Prime Numbers

Prime numbers are integral to this exercise's output. They are numbers greater than 1 that have no divisors other than 1 and itself. Essentially, a prime number is not divisible by any other number, which makes them survive the filtering of multiples.

By iterating over numbers 2 through 100 and filtering out multiples of each, the list is effectively reduced to numbers that include primes. Through this operation, you retain numbers that do not have divisors in the given range. This process resembles finding prime numbers using the Sieve of Eratosthenes, an ancient algorithm. Although the exercise stops at 100, continuing this logic could help build more expansive candidate lists of prime numbers.

Understanding prime numbers and applying filtering logic through algorithms reinforces concepts of divisibility and aids in recognizing efficient methods for identifying prime numbers in programming.